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We propose a phase-field model of dynamic fracture based on theAmbrosio--Tortorelli's approximation, which takes into account dissipativeeffects due to the speed of the crack tips. By adapting the time discretizationscheme contained in [C.J. Larsen, C. Ortner, and E. S\"uli, Math. ModelsMethods Appl. Sci. (2010)], we show the existence of a dynamic crack evolutionsatisfying an energy-dissipation balance, according to Griffith's criterion.Finally, we analyze the dynamic phase-field model of [B. Bourdin, C.J. Larsen,and C.L. Richardson, Int. J. Fracture (2011)] and [C.J. Larsen, IUTAM Symposiumon Variational Concepts with Applications to the Mechanics of Materials (2010)]with no dissipative terms.

Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation